On conflict-free colorings of cyclic polytopes and the girth conjecture for graphs

TL;DR

This paper investigates the conflict-free coloring of hypergraphs from cyclic polytopes, revealing connections to the Erdős girth conjecture and providing bounds in various dimensions.

Contribution

It introduces new bounds for conflict-free chromatic numbers in even dimensions and uncovers a surprising link to the Erdős girth conjecture.

Findings

Sharp asymptotic bounds in small even dimensions

Non-trivial bounds for general even dimensions

A novel connection to the Erdős girth conjecture

Abstract

We study the conflict-free chromatic number of hypergraphs derived from the family of facets of $d$-dimensional cyclic polytopes with $n$ vertices. While in odd dimensions $d$ the problem is easy, for even dimensions the problem becomes very difficult and exhibits interesting connections to extremal graph theory. We provide sharp asymptotic bounds for the conflict-free chromatic number in several small even dimensions and non-trivial upper and lower bounds for general even dimensions. The main purpose of this paper is revealing a surprising relation between conflict-free colorings and the celebrated Erd\H{o}s girth conjecture, opening new avenues for future research.